/* 

  Four digit palindrome in Picat.

  https://ai.ia.agh.edu.pl/_media/pl:dydaktyka:est:essential_thinking_2020-59.pdf
  """
  Four Digit Palindroma 
  - four digit palindrom: 1221, 7337, 2992,...
  - observe: 1221:11=111, 7337:11=667, 2992:11=272,...
  Hypothesis: Every four-digit palindrom number is divisible by 11.
  """
  page 26


  * go/0: Some of the 90 solutions for N=4:

     n = 4
     x = [1,0,0,1]
     x = [1,1,1,1]
     x = [1,2,2,1]
     x = [1,3,3,1]
     x = [1,4,4,1]
     x = [1,5,5,1]
     x = [1,6,6,1]
     x = [1,7,7,1]
     x = [1,8,8,1]
     x = [1,9,9,1]
     x = [2,0,0,2]
     x = [2,1,1,2]
     x = [2,2,2,2]
     x = [2,3,3,2]
     ...
     x = [9,3,3,9]
     x = [9,4,4,9]
     x = [9,5,5,9]
     x = [9,6,6,9]
     x = [9,7,7,9]
     x = [9,8,8,9]
     x = [9,9,9,9]


  * go2/0: Number of solutions when X mod 11 = 0, for N=2..14:

     2 = 9
     3 = 8
     4 = 90
     5 = 82
     6 = 900
     7 = 818
     8 = 9000
     9 = 8182
     10 = 90000
     11 = 81818
     12 = 900000
     13 = 818182
     14 = 9000000


  * go3/0: Number of solutions when X mod 11 != 0, for N=2..14.
    It seems that the hypothesis holds for all even N (see go3/0) but 
    not for odd N.

     2 = 0
     3 = 82
     4 = 0
     5 = 818
     6 = 0
     7 = 8182
     8 = 0
     9 = 81818
     10 = 0
     11 = 818182
     12 = 0
     13 = 8181818
     14 = 0
     15 = 81818182
     16 = 0


  This Picat model was created by Hakan Kjellerstrand, hakank@gmail.com
  See also my Picat page: http://www.hakank.org/picat/

*/

import cp.

main => go.


%
% Checking when mod 11 = 0: 90 solutions for n=4
%
go ?=>
  % N = 4,
  member(N,2..10),
  println(n=N),
  Mod #= 0,
  palindrome(N,Mod,X),
  solve(X),
  println(x=X),
  fail,
  nl.
go => true.


%
% Checking when mod 11 = 0 for N=2..15.
%
go2 ?=>
  member(N,2..14),
  Mod #= 0,
  Count = count_all(palindrome(N,Mod,_X)),
  println(N=Count),
  fail,
  nl.
go2 => true.


%
% Checking when mod 11 > 0: 0 solutions for n=4.
% See above for the general result, N=2..15.
%
go3 ?=>
  member(N,2..14),
  Mod #> 0,
  Count = count_all(palindrome(N,Mod,_X)),
  println(N=Count),
  fail,
  nl.
go3 => true.



palindrome(N,Mod,X) =>
  X = new_list(N),
  X :: 0..9,

  Y :: 10**(N-1)..10**N-1,

  to_num(X,10,Y),

  % is a palindrome
  foreach(I in 1..N div 2)
    X[I] #= X[N-I+1]
  end,

  Y mod 11 #= Mod,
  
  solve(X).


%
% converts a number Num to/from a list of integer List given a base Base
%
to_num(List, Base, Num) =>
   Len = length(List),
   Num #= sum([List[I]*Base**(Len-I) : I in 1..Len]).